Generalized B¨ acklund–Darboux transformations for Coxeter–Toda flows

c

2011 by Institut Mittag-Leffler. All rights reserved

Generalized B¨ acklund–Darboux transformations for Coxeter–Toda flows

from a cluster algebra perspective

by

Michael Gekhtman

University of Notre Dame Notre Dame, IN, U.S.A.

Michael Shapiro

Michigan State University East Lansing, MI, U.S.A.

Alek Vainshtein

University of Haifa Haifa, Mount Carmel, Israel

1. Introduction

This is the third in the series of papers in which we investigate Poisson geometry of di- rected networks. In [21] and [22], we studied Poisson structures associated with weighted directed networks in a disk and in an annulus. The study was motivated in part by Pois- son properties of cluster algebras. In fact, it was shown in [21] that if a universal Poisson bracket on the space of edge weights of a directed network in a disk satisfy an analogue of the Poisson–Lie property with respect to concatenation, then the Poisson structure induced by this bracket on the corresponding Grassmannian is compatible with the clus- ter algebra structure in the homogeneous coordinate ring of the Grassmannian. In this paper we deal with an example that ties together objects and concepts from the theory of cluster algebras and directed networks with the theory of integrable systems.

Integrable systems in question are the Toda flows on GL_n. These are commuting Hamiltonian flows generated by conjugation-invariant functions on GL_n with respect to the standard Poisson–Lie structure. Toda flows (also known ascharacteristic Hamilton- ian systems [30]) are defined for an arbitrary standard semisimple Poisson–Lie group, but we will concentrate on the GL_n case, where as a maximal algebraically indepen- dent family of conjugation-invariant functions one can chooseF_k: GL_n3X7!(1/k) trX^k, k=1, ..., n−1. The equation of motion generated byF_k has the Lax form

d dtX=

X,−1

2(π+(X^k)−π−(X^k))

, (1.1)

whereπ+(A) andπ−(A) denote strictly upper and lower parts of a matrixA.

Any double Bruhat cell G^u,v, u, v∈S_n, is a regular Poisson submanifold in GL_n invariant under the right and left multiplication by elements of the maximal torus (the subgroup of diagonal matrices) H⊂GL_n. In particular, G^u,v is invariant under the conjugation by elements of H. The standard Poisson–Lie structure is also invariant under the conjugation action ofHon GLn. This means that Toda flows defined by (1.1) induce commuting Hamiltonian flows onG^u,v/H, whereHacts onG^u,vby conjugation.

In the case when v=u⁻¹=(n 1 2 ... n−1), G^u,v consists of tridiagonal matrices with non-zero off-diagonal entries, andG^u,v/H can be conveniently described as the set Jac ofJacobi matrices of the form

L=



















b₁ 1 0 ... 0

a₁ b₂ 1 ... 0

... ... ... ... ...

0 ... an−2 bn−1 1 0 ... 0 an−1 bn



















, a₁... a_n−16= 0, detL6= 0. (1.2)

The Lax equations (1.1) then become the equations of thefinite non-periodic Toda hier- archy

d

dtL= [L, π−(L^k)],

the first of which, corresponding tok=1, is the celebratedToda lattice d

dta_j=a_j(b_j+1−b_j), j= 1, ..., n−1, d

dtbj= (aj−a_j−1), j= 1, ..., n,

with the boundary conditionsa₀=a_n=0. Recall that detLis a Casimir function for the standard Poisson–Lie bracket. The level sets of the function detLfoliate Jac into 2(n−1)- dimensional symplectic manifolds, and the Toda hierarchy defines a completely integrable system on every symplectic leaf. Note that although Toda flows on an arbitrary double Bruhat cell G^u,v can be exactly solved via the so-calledfactorization method (see, e.g.

[31]), in most cases the dimension of symplectic leaves inG^u,v/Hexceeds 2(n−1), which means that conjugation-invariant functions do not form a Poisson commuting family rich enough to ensure Liouville complete integrability.

An important role in the study of Toda flows is played by theWeyl function m(λ) =m(λ;X) = ((λ1−X)⁻¹e1, e1) =q(λ)

p(λ), (1.3)

wherep(λ) is the characteristic polynomial ofX andq(λ) is the characteristic polynomial of the (n−1)×(n−1) submatrix of X formed by deleting the first row and column (see, e.g., [5], [7] and [28]). Differential equations that describe the evolution of m(λ;X) induced by Toda flows do not depend on the initial value X(0) and are easy to solve:

though non-linear, they are also induced by linear differential equations with constant coefficients on the space

M(λ) =Q(λ)

P(λ): degP=n, degQ=n−1,P andQare coprime,P(0)6= 0

(1.4) by the mapM(λ)7!m(λ)=−M(−λ)/H0, whereH0=limλ!∞λM(λ)6=0.

It is easy to see thatm(λ;X) is invariant under the action ofH onG^u,v by conju- gation. Thus we have a map fromG^u,v/Hinto the space

W_n=

m(λ) =q(λ)

p(λ): degp=n, degq=n−1,pandqare monic and coprime,p(0)6= 0

. In the tridiagonal case, this map, sometimes called theMoser map, is invertible: it is a classical result in the theory of moment problems that matrix entries of an element in Jac can be restored from its Weyl functionm(λ;X) via determinantal formulas for matrix entries of X in terms of Hankel determinants built from the coefficients of the Laurent expansion of m(λ;X). These formulas go back to the work of Stieltjes on continuous fractions [33] (see, e.g. [1] for details).

In this paper, we study double Bruhat cellsG^u,v that share common features with the tridiagonal case:

(i) the Toda hierarchy defines a completely integrable system on level sets of the determinant inG^u,v/H, and

(ii) the Moser mapmu,v:G^u,v/H!Wndefined in the same way as in the tridiagonal case is invertible.

We will see that double Bruhat cellsG^u,v associated with any pair of Coxeter ele- mentsu, v∈Sn enjoy these properties. (Recall that a Coxeter element inSn is a product ofn−1 distinct elementary transpositions.) Double Bruhat cells of this kind has previ- ously appeared (for an arbitrary simple Lie group) in [23] in the context of integrable systems, and in [3] and [36] in connection with cluster algebras of finite type. We will call any such double Bruhat cell aCoxeter double Bruhat cell. Integrable equations induced onG^u,v/Hby Toda flows will be calledCoxeter–Toda lattices. This term was first used in [23] in the caseu=v for an arbitrary simple Lie group, which generalizes therelativistic Toda latticethat corresponds to the choiceu=v=s_n−1... s₁in GL_n. In [12] and [13], the corresponding integrable systems forv=s_n−1... s₁ and an arbitrary Coxeter element u were calledelementary Toda lattices. In the latter case, G^u,v/H can be described as a

subset of Hessenberg matrices subject to certain rank conditions on submatrices. The tridiagonal case corresponds to the choicev=s_n−1... s1andu=s1... s_n−1.

Since Coxeter–Toda flows associated with different choices of (u, v) lead to the same evolution of the Weyl function, and the corresponding Moser maps are invertible, one can construct transformations between differentG^u,v/Hthat preserve the corresponding Coxeter–Toda flows and thus serve as generalized B¨acklund–Darboux transformations between them.

Our goal is to describe these transformations from the cluster algebra point of view.

To this end, we construct a cluster algebra of rank 2n−2 associated with an extension of the space (1.4)

Rn= Q(λ)

P(λ): degP=n, degQ < n,P andQare coprime,P(0)6= 0

.

(Note thatWnis embedded intoRnas a codimension-1 subspace.) Distinguished clusters xu,vin this algebra correspond to Coxeter double Bruhat cells, and are formed by certain collections of Hankel determinants built out of coefficients of the Laurent expansion of an element inRn. Sequences of cluster transformations connecting these distinguished clusters are then used as the main ingredient in the construction of generalized B¨acklund–

Darboux transformations.

The insight necessary to implement this construction is drawn from two sources:

(i) the procedure for the inversion of the Moser map, which can be viewed as a generalization of the inverse moment problem, and

(ii) interpretation of functions inRnas boundary measurement functions associated with a particular kind of networks in an annulus.

Before discussing the organization of the paper, we would like to make two re- marks. First, birational transformations betweenG^{u,sn−1...s1}/H and G^{u0,sn−1...s1}/H for two different Coxeter elements uand u⁰ which serve as generalized B¨acklund–Darboux transformations between the corresponding elementary Toda lattices were first studied in [12]. Second, a cluster algebra closely related to the one we considered here recently appeared in [25] and was subject of a detailed combinatorial study in [10], where cluster mutations along the edges of a certain subgraph of its exchange graph were shown to describe an evolution of anAn-type Q-system—a discrete evolution arising in the anal- ysis of the XXX-model, which is an example of a quantum integrable model. In [10], solutions of the Q-system are represented as Hankel determinants built from coefficients of a certain generating function, which turns out to be rational and can be represented as a matrix element of a resolvent of an appropriate linear operator.

The paper is organized as follows.

In§2 we go over the necessary background information on double Bruhat cells, Toda flows, cluster algebras, networks and associated Poisson structures. We then proceed, in

§3, to describe a parametrization of a Coxeter double Bruhat cell. This is a particular case of the Berenstein–Fomin–Zelevinsky parametrization [2], [15]: for a generic element X inG^u,v, we consider a factorization ofX into elementary bidiagonal factors consistent with the Gauss factorization ofX, that isX=X−X0X+, whereX0is the diagonal matrix diag(d1, ..., dn), X+ is the product ofn−1 elementary upper bidiagonal factors E_i⁺(c⁺_i), i=1, ..., n−1, with the order of factors in the product prescribed by v, and X− is the product ofn−1 elementary lower bidiagonal factorsE⁻_i(c⁻_i),i=1, ..., n−1, with the order of factors in the product prescribed byu. We also give an intrinsic characterization of a double Bruhat cell.

Elements G^u,v/H are parametrized by d_i and c_i=c⁺_ic⁻_i, i=1, ..., n−1. In §4 we show that these parameters can be restored as monomial expressions in terms of an appropriately chosen collection of Hankel determinants built from the coefficients of the Laurent expansion of the Weyl function m(λ). (In [14], a similar inverse problem was solved for the case v=s_n−1... s₁, u arbitrary.) Both the choice of Hankel determinants and exponents entering monomial expressions ford_i andc_i are uniquely determined by the pair (u, v).

In §5, the map X7!m(λ;X) is given a combinatorial interpretation in terms of weighted directed planar networks. To an elementary bidiagonal factorization ofX∈G^u,v there corresponds a networkN_u,v in a square (disk) withn sources located on one side of the square and n sinks located at the opposite side, both numbered bottom to top (see, e.g. [11], [15] and [16]). By gluing opposite sides of the square containing sinks and sources in such a way that each sink is glued to the corresponding source and adding two additional edges, one incoming and one outgoing, one obtains a weighted directed network in an annulus (the outer and inner boundary circles of the annulus are formed by the remaining two sides of the square). Networks in an annulus were studied in [22]. The network we just described,N_u,v , has one sink and one source on the outer boundary of an annulus and, according to [22], the boundary measurement corresponding to this network is a rational functionM(λ) in an auxiliary parameterλ. We show that−M(−λ) is equal tom(λ;X) times the product of weights of the incoming and outgoing edges inN_u,v .

The determinantal formulas for the inverse of the Moser map are homogeneous of degree zero with respect to the coefficients of the Laurent expansion, and therefore the same formulas applied to M(λ) also recover ci and di. Thus, we can define a map

%u,v: (C^∗)²ⁿ!G^u,v/H in such a way that the through map

G^u,v/H ^mu,v //W_n  //R_n ^xu,v //(C^∗)^{2n %u,v} //G^u,v/H is the identity map.

In the remainder of§5, we use the combinatorial data determined by the pair (u, v) (or, in a more transparent way, by the corresponding networkN_u,v ) to construct a cluster algebraA=Au,v, with the (slightly modified) collectionxu,vserving as the initial cluster.

The matrixBu,vthat determines cluster transformations for the initial cluster is closely related to the incidence matrix of the graph dual to N_u,v . To constructAu,v, we start with the Poisson structure induced on boundary measurement functions by a so-called standard Poisson bracket on the space of face weights associated withN_u,v (this bracket is a particular case of the general construction for networks in the annulus given in [22]).

Initial cluster variables, viewed as functions on Rn, form a coordinate system in which this Poisson structure takes a particular simple form: the Poisson bracket of logarithms of any two functions in the family is constant. This allows us to follow the strategy from [20] to construct Au,v as a cluster algebra compatible with this Poisson bracket.

We then show that Au,v does not depend on the choice of Coxeter elements u and v, that is, that for any (u⁰, v⁰) the initial seed ofAu⁰,v⁰ is a seed in the cluster algebraAu,v. Therefore, the change of coordinatesT_u,v^u0,v0:x_u,v7!x_u⁰_,v⁰ is accomplished by a sequence of cluster transformations. Moreover, the ring of regular functions onR_ncoincides with the localization of the complex form ofAwith respect to the stable variables. We complete

§5 with the discussion of the interplay between our results and those of [10]. In particular, we provide an alternative proof for one of the main results of [10] concerning the Laurent positivity of the solutions of Q-systems.

In the final section, we interpret generalized B¨acklund–Darboux transformations between Coxeter–Toda lattices corresponding to different pairs of Coxeter elements in terms of the cluster algebraAby observing that the map

σ^u_u,v^0,v0=%u⁰,v⁰T_u,v^u0,v0τu,v:G^u,v/H−!G^u0,v0/H, (1.5) with τ_u,v being the right inverse of %_u,v, preserves flows generated by conjugation- invariant functions and makes the diagram

G^u,v/H ^σ

u0,v0

u,v //

mu,v

##G

GG GG GG

GG G^u0,v0/H

m_u0,v0

{{vvvvvvvvv

Wn

commutative. We obtain explicit formulas for σ_u,v^u0,v0 and, as a nice application, present formulas that transform solutions of the usual Toda lattice into solutions of the rel- ativistic one. Besides, we explain how one represents generalized B¨acklund–Darboux transformations as equivalent transformations of the networkN_u,v . Finally we show that classical Darboux transformations are also related to cluster algebra transformations via a formula similar to (1.5).

2. Preliminaries

In this section we collect the necessary background information on double Bruhat cells, Toda flows and directed networks on surfaces. Though notions and results that we will need on the first two subjects can be as easily stated for an arbitrary semisimple group, we will limit ourselves to the GLn case.

2.1. Double Bruhat cells

Let b+, n+, b− and n− be the algebras of upper triangular, strictly upper triangular, lower triangular and strictly lower triangular matrices, respectively.

The connected subgroups that correspond tob+,b−,n+and n− will be denoted by B+,B−,N+andN−, respectively. We denote byHthe maximal torus (the subgroup of diagonal matrices) in GL_n.

Everyξ∈gl_n can be uniquely decomposed into ξ=ξ−+ξ0+ξ+,

whereξ+∈n+, ξ−∈n− and ξ0 is diagonal. Consequently, for every X in an open Zariski dense subset of GLn there exists a uniqueGauss factorization

X=X−X₀X+, X+∈N+, X−∈N−, X₀∈H.

Let si, i∈[1, n−1], denote the elementary transposition (i, i+1) in the symmetric group Sn. A reduced decomposition of an element w∈Sn is a representation of w as a product w=si₁... si_l of the smallest possible length. A reduced decomposition is not unique, but the number l depends only onw and is called thelength of wand denoted byl(w). The sequence of indicesi=(i1, ..., il) corresponding to a given reduced decom- position ofwis called areduced wordforw. The unique element ofSn of maximal length (also calledthe longest element ofSn) is denoted byw0.

We will also need need a notion of a reduced word for an ordered pair (u, v) of elements in Sn. It is defined as follows: if (i1, ..., il(u)) is a reduced word for u and (i⁰₁, ..., i⁰_l(v)) is a reduced word forv, then any shuffle of the sequences (i1, ..., il(u)) and (−i⁰₁, ...,−i⁰_l(v)) is called a reduced word for (u, v).

Let us fix an embedding ofSn into GLn and denote the representative ofw∈Sn in GL_n by the same letter w. TheBruhat decompositions of GL_n with respect to B+ and B− are defined, respectively, by

GLn= [

u∈Sn

B+uB+ and GLn= [

v∈Sn

B−vB−.

For anyu, v∈Sn, thedouble Bruhat cell is defined as G^u,v=B+uB+∩B−vB−.

According to [15], the varietyG^u,v is biregularly isomorphic to a Zariski open sub- set of Cl(u)+l(v)+n. A corresponding birational map from Cl(u)+l(v)+n to G^u,v can be constructed quite explicitly, though not in a unique way. Namely, fix a reduced wordi for the pair (u, v) and consider, in addition, a sequencek=(k₁, ..., k_n) obtained as an ar- bitrary rearrangement of the numbers√

−1,2√

−1, ..., n√

−1. Letj=(j1, ..., jl(u)+l(v)+n) be a shuffle ofkandi; we set

θ(j_l) =







+, ifj_l>0,

−, ifj_l<0, 0, ifj_l∈k.

Denote byeij an elementaryn×nmatrix (δiαδjβ)ⁿ_α,β=1. Fort∈C,i, j∈[1, n−1] and k∈[1, n], let

E_i⁻(t) =1+tei+1i, E_j⁺(t) =1+tej j+1 and E⁰_k(t) =1+(t−1)ekk. (2.1) Then the mapXj:Cl(u)+l(v)+n!G^u,v can be defined by

X_j(t) =

l(u)+l(v)+n

Y

q=1

E_|j^θ(jq)

q| (t_q). (2.2)

The parameterst1, ..., tl(u)+l(v)+n constitutingtare calledfactorization parameters. Ex- plicit formulas for the inverse of the map (2.2) in terms of the so-calledtwisted generalized minors were found in [15].

2.2. Toda flows

Next, we review the basic facts about the Toda flows on GLn.

Recall that the standard Poisson–Lie structure on GLn is given by

{f1, f2}SL_n(X) =¹₂(R(∇f1(X)X),∇f2(X)X)−¹₂(R(X∇f1(X)), X∇f2(X)), where (·,·) denotes thetrace-form, ∇is the gradient defined with respect to the trace- form, andR:gl_n!gl_n is the standardR-matrix given by

R(ξ) =ξ+−ξ−= (sign(j−i)ξ_ij)ⁿ_i,j=1.

Double Bruhat cells are regular Poisson submanifolds of GLn equipped with the standard Poisson–Lie structure (see [26], [30] and [35]). Furthermore,

(i) any symplectic leaf of GLn is of the form S^u,va, where S^u,v⊂G^u,v is a certain distinguished symplectic leaf andais an element of the Cartan subgroup, and

(ii) the dimension of symplectic leaves inG^u,vequalsl(u)+l(v)+corank(uv⁻¹−Id), see [26] and [30].

Conjugation-invariant functions on GLnform a Poisson-commuting family (see, e.g., [31]). Any such functionF generates a Hamiltonian flow described by theLax equation

dX dt =

X,−1

2R(X∇F(X))

. (2.3)

The resulting family of equations is calledthe hierarchy of Toda flows (in [30], the term characteristic Hamiltonian systems is used). If one choosesF(X)=Fk(X)=(1/k) trX^k, then equation (2.3) becomes (1.1). The functionsF1, ..., F_n−1 form a maximal family of algebraically independent conjugation-invariant functions on GLn.

For an element h∈GLn, denote by Ch the action of h on GLn by conjugation:

Ch(X)=hXh⁻¹. For any smooth functionf on GLn we have

∇(fCh) = Ad_h⁻¹(∇f).

Furthermore, ifhbelongs toH, then it is easy to see that R(Ad_h−1(ξ)) = Ad_h−1(R(ξ))

for any ξ∈gl_n. Together, these observations imply that for any h∈H and any pair of smooth functionsf1 andf2 on GLn,

{f1Ch, f2Ch}={f1, f2}Ch.

In other words, the action ofH on GLn by conjugation is Poisson with respect to the standard Poisson–Lie structure. Since the action preserves double Bruhat cells, the standard Poisson–Lie structure induces a Poisson structure on G^u,v/H, and the Toda hierarchy induces the family of commuting Hamiltonian flows onG^u,v/H.

Remark 2.1. (i) The Lax equation (2.3) can be solved explicitly via thefactorization method [31], which we will not review here.

(ii) Written in terms of matrix entries, equations (2.3) have exactly the same form as equations of the Toda hierarchy on gl_n, where the relevant Poisson structure is the Lie–Poisson structure associated with theR-matrix Lie bracket

[ξ, η]_R=¹₂([R(ξ), η]+[ξ, R(η)]).

In fact, viewed as equations on the algebra ofn×nmatrices, the Toda hierarchy becomes a family of biHamiltonian flows with compatible linear and quadratic Poisson brackets given by Lie–Poisson and the extension of the Poisson–Lie brackets, respectively. How- ever, we will not need the linear Poisson structure in the current paper.

2.3. Cluster algebras and compatible Poisson brackets

First, we recall the basics of cluster algebras of geometric type. The definition that we present below is not the most general one, see, e.g., [3] and [17] for a detailed exposition.

Thecoefficient group Pis a free multiplicative abelian group of finite rank mwith generators g1, ..., gm. An ambient field is the field F of rational functions in n inde- pendent variables with coefficients in the field of fractions of the integer group ring ZP=Z[g₁^±1, ..., g^±1_m] (here we writex^±1 instead ofx, x⁻¹). It is convenient to think ofF as of the field of rational functions in n+m independent variables with rational coeffi- cients.

Aseed(ofgeometric type) inFis a pair Σ=(x, B), wherex=(x₁, ..., x_n+m),x₁, ..., x_n is a transcendence basis ofFover the field of fractions ofZP,x_n+i=g_ifori∈[1, m], andB is ann×(n+m) integer matrix whose principal part (that is, then×nsubmatrix formed by the columns 1, ..., n) is skew-symmetric. The (n+m)-tuplex is called a cluster, its elementsx₁, ..., x_n are calledcluster variables, and x_n+1, ..., x_n+m arestable variables.

Given a seed as above, thecluster transformationin direction k∈[1, n] is defined by x7−!xk= (x\{xk})∪{¯xk},

where the new cluster variable ¯xk is given by theexchange relation xkx¯k= Y

16i6n+m b_ki>0

x^b_i^ki+ Y

16i6n+m b_ki<0

x^−b_i ^ki; (2.4)

here, as usual, the product over the empty set is assumed to be equal to 1.

We say thatBis obtained fromB by amatrix mutation in directionk if

¯bij=

−bij, ifi=korj=k, b_ij+¹₂(|b_ik|b_kj+b_ik|b_kj|), otherwise.

Given a seed Σ=(x, B), we say that a seedΣ=(¯x,B) is adjacentto Σ (in directionk) if ¯xis obtained fromxandBis obtained fromBby a cluster transformation and a matrix mutation, respectively, in directionk. Two seeds aremutation equivalent if they can be connected by a sequence of pairwise adjacent seeds. The cluster algebra (of geometric

type) A=A(B) associated with Σ is the ZP-subalgebra of F generated by all cluster variables in all seeds mutation equivalent to Σ. The complex form ofAis defined as A tensored byCand is denoted by A_C.

LetV be a Zariski open subset inC^n+m, andA_C be the complex form of a cluster algebra of geometric type. We assume that the variables in some extended cluster are identified with a set of algebraically independent rational functions on V. This allows us to identify cluster variables in any cluster with rational functions onV as well, and thus to considerA_Cas a subalgebra of the fieldC(V) of rational functions onV. Finally, we denote byA^V_C the localization ofA_Cwith respect to the stable variables that do not vanish onV.

Proposition 2.1. Let V and A as above satisfy the following conditions:

(i) each regular function on V belongs to A^V_C;

(ii) there exists a cluster x=(x1, ..., xn+m) in A_C consisting of algebraically inde- pendent functions regular on V;

(iii) any cluster variable x¯_k, k∈[1, n], obtained by the cluster transformation (2.4) applied to x is regular on V.

Then A^V_C is isomorphic to the ring O(V)of regular functions on V.

Proof. All we have to prove is that any element inA^V_C is a regular function on V. The proof follows the proof of a similar statement for double Bruhat cells in [37] and consists of three steps.

Lemma2.1. Let z=(z₁, ..., z_n+m)be an arbitrary cluster in A_C. If a Laurent mono- mial M=z₁^d1... z_n+m^dn+m is regular on V then d_i>0 for i∈[1, n].

Proof. Indeed, assume that d_k<0 for some k∈[1, n] and consider the cluster z_k. By (2.4), M can be rewritten asM=M₁z¯_k^−dk/P^−dk, where M₁ is a Laurent monomial in the common variables ofzandz_k, andP is the binomial (in the same variables) that appears in the right-hand side of (2.4). By condition (i) and the Laurent phenomenon (Theorem 3.1 in [17]),M can be written as a Laurent polynomial in the variables ofz_k. Equating two expressions forM, we see thatP^−dk times a polynomial in the variables of z_k equals a Laurent monomial in the same variables. This contradicts the algebraic independence of the variables inz_k, which follows from the algebraic independence of the variables inx.

Lemma 2.2. Let z be a cluster variable in an arbitrary cluster z, and assume that zis a regular function on V. Then z is irreducible in the ring of regular functions onV. Proof. Without loss of generality, assume thatz_n+1=x_n+1, ..., z_n+m⁰=x_n+m⁰ do not vanish onV, and z_n+m⁰₊₁=x_n+m⁰₊₁, ..., z_n+m=x_n+m may vanish onV. Moreover, as-

sume on the contrary that z=f g, where f and g are non-invertible regular functions onV. By condition (i) and Proposition 11.2 of [18], both f and g are Laurent polyno- mials inz1, ..., zn+m⁰ whose coefficients are polynomials inzn+m⁰+1, ..., zn+m. Applying the same argument as in the proof of Lemma 2.1, we see that bothf andg are, in fact, Laurent monomials in z1, ..., zn+m and that zn+1, ..., zn+m⁰ enter both f and g with a non-negative degree. Moreover, by Lemma 2.1, each cluster variablez1, ..., znenters both f andg with a non-negative degree. This can only happen if one off andgis invertible in O(V), a contradiction.

Denote by U₀⊂V the locus of allt∈V such thatx_i(t)6=0 for all i∈[1, n]. Besides, denote byU_k⊂V the locus of allt∈V such thatx_i(t)6=0 for alli∈[1, n]\kand ¯x_k(t)6=0.

Lemma 2.3. Let U=Sn

i=0Ui. Then codimV\U>2.

Proof. This follows immediately from Lemma 2.2 and conditions (ii) and (iii).

Assume that there exists f∈A^V

C which is not regular on V. Recall that V is the complement of a finite union of irreducible hypersurfaces D_i in C^n+m. Therefore, the divisor of the poles off has codimension 1 in C^n+m. Since f is not regular onV, this latter divisor does not lie entirely in the union of D_i, and hence its intersection with V has codimension 1 inV. Therefore, by Lemma 2.3, it intersects U non-trivially. To complete the proof, note that by Proposition 11.2 of [18], any function inA^V

C is regular onU, a contradiction.

Let{ ·,· }be a Poisson bracket on the ambient fieldF. We say that it iscompatible with the cluster algebra A(B) if, for any clusterx=(x1, ..., xn+m), one has

{xi, xj}=ωijxixj,

where ω_ij∈Z are constants for all i, j∈[1, n+m]. The matrix Ω^x=(ω_ij) is called the coefficient matrix of{ ·,· }(in the basis x); clearly, Ω^x is skew-symmetric. A complete description of Poisson brackets compatible with A(B) in the case where rankB=n is given in [20].

2.4. Networks on surfaces with boundaries

LetSbe a disk withc>0 holes, so that its boundary∂Shasc+1 connected components, and let G=(V, E) be a directed graph embedded in S with the vertex set V and the edge setE. Exactlyrof its vertices are located on the boundary ∂S. They are denoted b₁, ..., b_r and calledboundary vertices. Each boundary vertex is labeled as a source or a sink. Asource is a vertex with exactly one outgoing edge and no incoming edges. Sinks

are defined in the same way, with the direction of the single edge reversed. The number of sources is denoted bynand the number of sinks bym=r−n. All the internal vertices of Ghave degree 3 and are of two types: either they have exactly one incoming edge, or exactly one outgoing edge. The vertices of the first type are called (and shown on figures)white, those of the second type,black.

A pair (v, e),v∈V,e∈E, is called aflag ifvis an endpoint ofe. To each flag (v, e) we assign an independent variable xv,e. Let u and v be two endpoints ofe. The edge weight weis defined bywe=xv,exu,e. Aperfect network N=(G, w, %1, ..., %c) is obtained fromG, weighted as above, by addingcnon-intersecting oriented curves%i (calledcuts) in such a way that cutting S along all %i makes it into a disk (note that the endpoints of each cut belong to distinct connected components of∂S). The points of thespace of edge weights EN=(R\{0})^|E|(or (C\{0})^|E|) can be considered as copies of the graphG with edges weighted by non-zero numbers obtained by specializing the variablesxv,e to non-zero values.

Assign an independent variableλ_i to each cut%_i. Theweight of a pathP between two boundary vertices is defined as the product of the weights of all edges constituting the path times a Laurent monomial in λ_i. Each intersection point of P with%_i contributes to this monomialλ_i if the oriented tangents toP and%_i at this point form a positively oriented basis, andλ⁻¹_i otherwise (assuming that all intersection points are transversal).

Besides, the sign of the monomial is defined via the rotation number of a certain closed curve built from P itself, and cuts and arcs of ∂S. For a detailed description of the corresponding constructions, see [21] and [29] in the case c=0 (networks in a disk, no cuts needed, the path weight is a signed product of the edge weights) and [22] in the casec=1 (networks in an annulus, one cut%and one additional independent variableλ involved, the path weight is a signed product of the edge weights times an integer power ofλ). Theboundary measurement between a sourcebi and a sinkbj is then defined as the sum of path weights over all (not necessary simple) paths frombito bj. It is proved in the above cited papers that a boundary measurement is a rational function in the weights of edges (in the case of the disk) or in the weights of edges andλ(in the case of the annulus).

Boundary measurements are organized in the boundary measurement matrix, and thus give rise to theboundary measurement mapfromEN to the space ofn×mmatrices (forc=0), or the space of n×m rational matrix functions (forc=1). The gauge group acts onE_N as follows: for any internal vertexvofN and any Laurent monomialLin the weightsw_eofN, the weights of all edges leavingv are multiplied byL, and the weights of all edges entering v are multiplied by L⁻¹. Clearly, the weights of paths between boundary vertices, and hence boundary measurements, are preserved under this action.

Therefore, the boundary measurement map can be factorized through the space FN

defined as the quotient ofEN by the action of the gauge group. In [22] we explained that FN can be identified with the relative cohomology groupH¹(G, G∩∂S) with coefficients in the multiplicative group of non-zero real numbers. This gives rise to the representation

FN=H¹(G∪∂S)/H¹(∂S)⊕H⁰(∂S)/H⁰(G∪∂S) =F_N^f⊕F_N^t.

The space F_N^f can be described as follows. The graph G divides S into a finite number of connected components called faces. The boundary of each face consists of edges of Gand, possibly, of several arcs of∂S. A face is called bounded if its boundary contains only edges of Gand unbounded otherwise. Given a facef, we define its face weight y_f as the function on E_N which assigns to the edge weightsw_e,e∈E, the value

y_f= Y

e∈∂f

w^γ_e^e,

where γe=1 if the direction ofe is compatible with the counterclockwise orientation of the boundary ∂f and γe=−1 otherwise. Face weights are invariant under the gauge group action, and hence are functions onF_N^f, and, moreover, form a basis in the space of such functions.

In [21] and [22] we studied the ways to turn the space of edge weights into a Poisson manifold by considering Poisson brackets on the space of flag variables satisfying certain natural conditions. We proved that all such Poisson brackets onEN form a 6-parameter family, and that this family gives rise to a 2-parameter family of Poisson brackets on FN. In what follows we are interested in a specific member of the latter family (ob- tained by setting α=¹₂ and β=−¹₂ in the notation of [21] and [22]). For reasons that will be explained later, we call this bracket the standard Poisson bracket on FN. The corresponding 4-parameter family of Poisson brackets onEN is called standard as well.

Given a perfect networkN as above, define thedirected dual networkN^∗=(G^∗, w^∗) as follows. Vertices of G^∗ are the faces ofN. Edges of G^∗ correspond to the edges of N which connect either two internal vertices of different colors, or an internal vertex with a boundary vertex; note that there might be several edges between the same pair of vertices in G^∗. An edge e^∗ of G^∗ corresponding to e is directed in such a way that the white endpoint of e(if it exists) lies to the left ofe^∗ and the black endpoint ofe (if it exists) lies to the right of e. The weight w^∗(e^∗) equals 1 if both endpoints of e are internal vertices, and ¹₂ if one of the endpoints ofeis a boundary vertex.

Proposition 2.2. The restriction of the standard Poisson bracket on FN to the space F_N^f is given by

{yf, yf⁰}=

X

e^∗:f!f⁰

w^∗(e^∗)− X

e^∗:f⁰!f

w^∗(e^∗)

yfyf⁰.

1 1

2 2

n n

d1

d2

dn

1 1

i i

i+1 i+1

n n

l

1 1

j j

j+1 j+1

n n

u

(a) (b) (c)

Figure 1. Three building blocks used in matrix factorization.

For networks in a disk, the above proposition is a special case of [21, Lemma 5.3].

For other surfaces the proof is literally the same.

In what follows, we will deal with networks of two kinds: acyclic networks in a disk with the same number of non-alternating sources and sinks, and networks in an annulus obtained from the networks of the first kind by a certain construction, to be described below.

In the former case we assume thatnsources are numbered clockwise and are followed bynsinks numbered counterclockwise. The weight of a path in this case is exactly the product of edge weights involved. The boundary measurements are organized into an×n matrixX in such a way thatX_ij is the boundary measurement between theith source and thejth sink. One can concatenate two networks of this kind by gluing the sinks of the former to the sources of the latter. IfX₁andX₂are the matrices associated with the two networks, then the matrix associated with their concatenation isX₁X₂. This fact can be used to visualize parametrization (2.2). Indeed, ann×ndiagonal matrix diag(d₁, ..., d_n) and elementary bidiagonal matrices E⁻_i(l) and E_j⁺(u) defined by (2.1) correspond to building blocks shown on Figure 1 (a), (b) and (c), respectively; all weights not shown explicitly are equal to 1. Note that building blocks themselves are not networks, since their edge weights do not comply with the rules introduced above. However, as we will see below, objects glued from building blocks comply with all the rules.

The concatenation ofn(n−1) building blocks of the second and the third types and one building block of the first type, in an appropriately chosen order and with each building block having its own non-trivial weights, describes a generic element of GL_n (see, e.g. [11]). The structure of the obtained network is given by Figure 2. Here and in what follows we use the gauge group action to decrease the number of parameters of networks in question. In particular, this network has 2n(n−1) internal vertices, and therefore, one can use the gauge group action to change the weights of 2n(n−1) edges to 1. It is convenient to choose these edges to be all the horizontal edges except for one middle edge in each horizontal chain. The weights on the remaining edges are Laurent

1 2 3 n−1 n

1 2 3 n−1 n

l₁₁ l₂₂ l_n−1,n−1

l₂₁ ln−2,1 l_n−1,2

ln−1,1

d₁ d₂ d₃ d_n−1

dn

un−1,1 un−2,1 u_n−1,2

u₂₁ u₁₁

u₂₂ u_n−1,n−1

Figure 2. A generic planar network; the weights of edges are Laurent monomials of the initial weights.

monomials in the initial weights of the network. For example, if the endpoints of an edge e belong to levels i and i+1, then u_e=w_ew_Pi+1/w_Pi, where P_i and P_i+1 are the horizontal paths from the endpoints ofeto the sinksiand i+1, respectively.

The following result, which is a special case of Theorem 4.1 from [21], explains why we call the bracket in consideration standard.

Theorem 2.1. For any network N as above with n sources and n sinks, the map from EN to the space of n×n matrices given by the boundary measurement matrix is Poisson with respect to any standard Poisson bracket on EN and the standard Sklyanin bracket on GLn.

Remark 2.2. Note that the definition of theR-matrixR_α,β in [21] contains a super- fluous factor ¹₂.

Networks in an annulus that we study in this paper are obtained from the above de- scribed networks in a disk by a gluing procedure described in detail in§5. These networks have one source and one sink, both lying on the same connected component of the bound- ary. The other connected component of the boundary does not carry boundary vertices, and hence for our networks H⁰(∂S)=H⁰(G∪∂S), which implies thatFN=F_N^f. There- fore, the standard Poisson bracket onFN is completely described by Proposition 2.2.

3. Coxeter double Bruhat cells

We start this section with describing a particular instance of the Berenstein–Fomin–

Zelevinsky parametrization [2], [15] in the case of Coxeter double Bruhat cells in GL_n. Lets_[p,q]=s_ps_p+1... s_q−1for 16p<q6nand recall that every Coxeter elementv∈Sn

can be written in the form

v=s_[ik−1,ik]... s_[i1,i2]s_[1,i1] (3.1) for some subset I={1=i₀<i₁<...<i_k=n}⊆[1, n]. Besides, define L={1=l₀<l₁<...<

l_n−k=n}by{l₁<...<l_n−k−1}=[1, n]\I.

Lemma3.1. Let v be given by (3.1). Then

v⁻¹=s_{[ln−k−1,ln−k]}... s_[l1,l2]s_[1,l1].

Proof. We use induction onn. Denote the right-hand side of the above relation by

¯

v. The indexn−1 belongs either toI or toL. In the latter case l_n−k−1=n−1, and we have

v=s_[ik−1,n]... s_[i1,i2]s_[1,i1]=s_{[ik−1,n−1]}... s_[i1,i2]s_[1,i1]s_n−1 and

¯

v=s_n−1s_{[ln−k−2,ln−k−1]}... s_[l1,l2]s_[1,l1].

Thenv=v⁰s_n−1and ¯v=s_n−1¯v⁰, wherev⁰ and ¯v⁰ are the Coxeter elements inS_n−1corre- sponding to the index setsI\{n}∪{n−1}andL\{n}, respectively, and hencevv¯=v⁰v¯⁰=1 by the induction hypothesis. Otherwise, ifn−1 belongs toI, we interchange the roles of v and ¯vand use the same argument.

Lemma3.2. The permutation matrix corresponding to a Coxeter element v is

˜ v=

k

X

j=1

e_ij−1ij+

n−k

X

j=1

e_ljlj−1.

Proof. We use the same inductive argument as in the proof of Lemma 3.1. Assuming thatn−1∈L, the relationv=v⁰s_n−1 and the induction hypothesis imply that

˜

v= (e1i₁+...+eik−1n−1+el₁1+...+e_n−1ln−k−2+enn)(e11+...+e_n−2n−2+e_{n n−1}+e_n−1n)

=e1i₁+...+eik−1n+el₁1+...+en−1ln−k−2+en n−1

as claimed.

Let now (u, v) be a pair of Coxeter elements and let I⁺={1 =i⁺₀< i⁺₁< ... < i⁺_k+=n}, I⁻={1 =i⁻₀< i⁻₁ < ... < i⁻_k−=n},

L⁺={1 =l⁺₀< l⁺₁< ... < l⁺_n−k+−1< l⁺_n−k+=n}, L⁻={1 =l⁻₀ < l₁⁻< ... < l⁻_n−k−−1< l⁻_n−k−=n}

(3.2)

be subsets of [1, n] corresponding tovandu⁻¹in the way just described. For a set of com- plex parametersc⁻₁, ..., c⁻_n−1;c⁺₁, ..., c⁺_n−1;d₁, ..., d_n, define matricesD=diag(d₁, ..., d_n),

C_j⁺=

i⁺_j−1

X

α=i⁺_j−1

c⁺_αe_α,α+1, j∈[1, k⁺], C_j⁻=

i⁻_j−1

X

α=i⁻_j−1

c⁻_αe_α+1,α, j∈[1, k⁻],

C_j⁺=

l⁺_j−1

X

α=l⁺_j−1

c⁺_αeα,α+1, j∈[1, n−k⁺], C_j⁻=

l⁻_j−1

X

α=l⁻_j−1

c⁻_αeα+1,α, j∈[1, n−k⁻].

(3.3)

Lemma 3.3. A generic element X∈G^u,v can be written as

X= (1−C₁⁻)⁻¹...(1−C_k⁻−)⁻¹D(1−C_k⁺+)⁻¹...(1−C₁⁺)⁻¹, (3.4) and its inverse can be factored as

X⁻¹= (1+C_n−k⁺ +)⁻¹...(1+C₁⁺)⁻¹D⁻¹(1+C₁⁻)⁻¹...(1+C_k⁻−)⁻¹. (3.5) Proof. It is easy to see that

(1−C_j⁺)⁻¹=E⁺

i⁺_j−1(c⁺

i⁺_j−1)... E⁺

i⁺_j−1(c⁺

i⁺_j−1), (1−C_j⁻)⁻¹=E⁻

i⁻_j−1(c⁻

i⁻_j−1)... E⁻

i⁻_j−1(c⁻

i⁻_j−1), (1+C_j⁺)⁻¹=E⁺

l⁺_j−1(−c⁺

l⁺_j−1)... E⁺

l⁺_j−1(−c⁺

l⁺_j−1), (1+C_j⁻)⁻¹=E⁻

l⁻_j−1(−c⁻_l−

j−1)... E⁻

l⁻_j−1(−c⁻_l− j−1

).

(3.6)

Then, by (2.2) and (3.1), a genericX∈G^u,v can be written as in (3.4). Next, the same reasoning as in the proof of Lemma 3.1 implies that

(1−C₁⁺)...(1−C_k⁺+) = ⁱ

+ k−1

Y

s=i⁺_k−1

E_s⁺(c⁺_s)

...

ⁱ

+ 1−1

Y

s=1

E_s⁺(c⁺_s) −1

= ^l

+ n−k−1

Y

s=l⁺_n−k−1

E_s⁺(−c⁺_s)

...

^l

+ 1−1

Y

s=1

E_s⁺(−c⁺_s)

= (1+C_n−k⁺ +)⁻¹...(1+C₁⁺)⁻¹, and, similarly,

(1−C_k⁻−)...(1−C₁⁻) = (1+C₁⁻)⁻¹...(1+C_k⁻−)⁻¹. Therefore,

X⁻¹= (1−C₁⁺)...(1−C_k⁺+)D⁻¹(1−C_k⁻−)...(1−C₁⁻)

= (1+C_n−k⁺ +)⁻¹...(1+C₁⁺)⁻¹D⁻¹(1+C₁⁻)⁻¹...(1+C_k⁻−)⁻¹.

The network Nu,v corresponding to the factorization (3.4) is obtained by the con- catenation (left to right) of 2n−1 building blocks (as depicted in Figure 1) corresponding to the elementary matrices

E⁻

i⁻₂−1(c⁻

i⁻₂−1), ..., E₁⁻(c⁻₁), E⁻

i⁻₃−1(c⁻

i⁻₃−1), ..., E⁻

i⁻₂(c⁻

i⁻₂), ..., E_n−1⁻ (c⁻_n−1), ..., E⁻

i⁻_{k− −1}(c⁻

i⁻_{k− −1}), D, E⁺

i⁺

k+−1

(c⁺

i⁺

k+−1

), ..., E_n−1⁺ (c⁺_n−1), ..., E_i⁺+

2

(c⁺_i+ 2

), ..., E_i⁺+ 3−1(c⁺_i+

3−1), E₁⁺(c⁺₁), ..., E_i⁺+ 2−1(c⁺_i+

2−1).